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**Unformatted text preview: **Qualifying Examination in Linear Algebra and Field Theory August, 2007 Answer as many questions as you can, but be sure to answer sufficiently many questions to comprise 90 points, with at least 30 points from each section. Write your answer to each problem on a separate sheet, placing the problem number in the upper right corner. Be sure to justify all your answers. Do not cite any result that reduces a proof to a triviality. Field Theory 1. [5,7,5] Let K be a field and let be an element of an algebraic closure of K . a. Let f ( X ) K [ X ] be the monic irreducible polynomial of over K and suppose deg f = n . Let L denote the splitting field of f ( X ) over K . Find an upper bound for [ L : K ] in terms of n . Prove that your answer is correct. b. Suppose [ K ( ) : K ] is odd. Prove that K ( 2 ) = K ( ). c. Let E denote the splitting field of X 5- 1 over Q . Find a primitive element for E , i.e., find a complex number so that E = Q ( ). Prove that your answer is correct....

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